Question

the graph of the normal curve
the diagram below shows a graph of a normal curve. the mean and the number of standard deviations from the mean are labeled below the graph. the red dashed lines correspond to the different values.
given that the mean is 63 and the standard deviation is 7, fill in the answer - boxes below the labels with their corresponding values.
-3 sd -2 sd -1 sd mean +1 sd +2 sd +3 sd

Explanation:

Step1: Calculate -3 standard - deviation value

The formula to find a value \(x\) in a normal distribution is \(x=\mu + z\sigma\), where \(\mu\) is the mean, \(z\) is the number of standard - deviations from the mean, and \(\sigma\) is the standard deviation. For \(z=-3\), \(\mu = 63\), and \(\sigma = 7\), we have \(x=63+(-3)\times7=63 - 21 = 42\).

Step2: Calculate -2 standard - deviation value

Using the same formula with \(z = - 2\), \(x=63+(-2)\times7=63-14 = 49\).

Step3: Calculate -1 standard - deviation value

For \(z=-1\), \(x=63+(-1)\times7=63 - 7=56\).

Step4: The mean value

The mean \(\mu = 63\).

Step5: Calculate +1 standard - deviation value

For \(z = 1\), \(x=63+1\times7=63 + 7=70\).

Step6: Calculate +2 standard - deviation value

For \(z = 2\), \(x=63+2\times7=63 + 14=77\).

Step7: Calculate +3 standard - deviation value

For \(z = 3\), \(x=63+3\times7=63 + 21=84\).

Answer:

42, 49, 56, 63, 70, 77, 84